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O-Level Differentiation and integration
What the O-Level syllabus expects for Differentiation and integration, and how to practise it.
What the syllabus expects
- Interpreting the derivative as a rate of change
- Using the standard notations dy/dx, d^2y/dx^2, f'(x), f''(x)
- Differentiating x^n for any rational n, plus sin x, cos x, tan x, e^x and ln x, along with their constant multiples, sums and differences
- Differentiating products of functions and quotients of functions
- Applying the Chain Rule
- Working with increasing and decreasing functions
- Distinguishing a maximum from a minimum with the second derivative test
- Applying differentiation to gradients, tangents and normals, to connected rates of change, and to maximum and minimum problems
- Treating integration as the inverse of differentiation
- Integrating x^n for any rational n, plus sin x, cos x, sec^2 x and e^x, together with their constant multiples, sums and differences
- Reading the definite integral as the area beneath a curve
- Computing the values of definite integrals
- Finding the area of a region enclosed by a curve and one or more lines
Scope: The area of a region lying between two curves is not included - Finding the area of regions that sit below the x-axis
- Applying differentiation and integration to a particle's straight-line motion involving displacement, velocity and acceleration
How it's examined
Questions on this topic most often ask you to find, determine, evaluate, solve. About 24% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (3 marks)
Show that d/dx(tan³2x) = 6sec⁴2x − 6sec²2x.
Show the worked answer
Let y = tan³(2x) = (tan 2x)³. Using the chain rule: dy/dx = 3(tan 2x)² · d/dx(tan 2x) = 3 tan²(2x) · (2 sec² 2x) = 6 tan²(2x) sec²(2x). Use the identity tan² θ = sec² θ − 1: = 6(sec² 2x − 1) sec² 2x = 6 sec⁴ 2x − 6 sec² 2x, as required.
Example 2 (3 marks)
Find the derivative with respect to x of ln(3x/(x² + 1)).
Show the worked answer
Use log laws first: ln(3x/(x² + 1)) = ln 3 + ln x - ln(x² + 1). Differentiate term by term: d/dx(ln 3) = 0, d/dx(ln x) = 1/x, d/dx(ln(x² + 1)) = 2x/(x² + 1). So dy/dx = 1/x - 2x/(x² + 1). Combining over a common denominator: = [(x² + 1) - 2x²]/[x(x² + 1)] = (1 - x²)/[x(x² + 1)].
Example 3 (3 marks)
For v = 3t² - 8t + 4, determine the minimum velocity of the particle.
Show the worked answer
v = 3t² - 8t + 4 dv/dt = 6t - 8 Minimum when dv/dt = 0: 6t - 8 = 0 => t = 4/3 v = 3(4/3)² - 8(4/3) + 4 = 16/3 - 32/3 + 12/3 = -4/3 Minimum velocity = -4/3 (approx -1.33)
More worked questions on this topic
- Hence, or otherwise, prove that x³ − 125 = −2x³ + 7x² + 16x − 5 has only one real root. (4 marks)
- With A = (1/4)x²(sin2θ + 2cosθ) and θ free to vary, find the value of θ that makes A stationary (3 marks)
- Given v = 3t² - 8t + 4, find the set of t-values for which this particle travels in a positive (3 marks)
- With v = 3t² - 8t + 4, work out the total path length covered by the particle over the first 3 (3 marks)
- Part of the curve y = √(2x + 5) is shown, with A the point on it where x = 2; the normal at A c (5 marks)
- For that open tank (base 4l by l, height h, outer surface area 5 m², volume V = (2/5)(5l − 4l³) (4 marks)
- Baking powder falls onto a surface at 2π cm³ s⁻¹, forming a right circular cone whose radius is (5 marks)
- A particle's acceleration is a = −2 cos(t/2) m/s², t seconds after leaving O. Sketch the accele (2 marks)
More O-Level Additional Maths (A-Maths) topics
Quadratic functions · Equations and inequalities · Surds · Polynomials and partial fractions · Binomial expansions · Exponential and logarithmic functions · all of O-Level Additional Maths (A-Maths)